Question

Is there any short proof for "an ellipse is the locus of all points where the distance to the first focus plus the distance to the second focus = 2a"

I've been trying to prove it myself, but the equation can't even fit on my paper anymore because of all the exponentiations I had to do, so I give up xD

Answer 1

shorter proof ?
what else can be a proof apart from the one that comes from the definition..

Answer 2

Well the standard equation of an ellipse:

\[\frac{(x-x _{1})^{2}}{a ^{2}} + \frac{(y-y _{1})^{2}}{b ^{2}} = 1\]

Doesn't relate anything to the foci.

So I tried going from this:

\[\sqrt{(x-(x _{1}-f))^{2}+(y-y _{1})^{2}}+\sqrt{(x-(x _{1}+f))^{2}+(y-y _{1})^{2}} = 2a\]

To the standard equation. But the way I tried doing it takes way too long

Answer 3

can you draw me your figure ?